Compromise Solutions in Multicriteria Optimization Kai-Simon - PowerPoint PPT Presentation

Introduction Definitions and Notations Basic Properties Approximation Compromise Solutions in Multicriteria Optimization Kai-Simon Goetzmann (Joint work with Christina B¨ using and Jannik Matuschke) OR 2011 – August 31 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Multicriteria Optimization Pareto optimal solutions: No objective can be improved without worsening another. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Multicriteria Optimization Pareto optimal solutions: No objective can be improved without worsening another. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Multicriteria Optimization Pareto optimal solutions: No objective can be improved without worsening another. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Avoiding too many solutions Scalarization: Maximize (weighted) sum of all objectives. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Avoiding too many solutions Scalarization: Maximize (weighted) sum of all objectives. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Avoiding too many solutions Scalarization: Maximize (weighted) sum of all objectives. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Avoiding too many solutions Scalarization: Maximize (weighted) sum of all objectives. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Avoiding too many solutions Scalarization: Maximize (weighted) sum of all objectives. f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Our Answer Compromise Solutions: Minimize distance to ideal point (w.r.t. some metric). f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Our Answer Compromise Solutions: Minimize distance to ideal point (w.r.t. some metric). f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Our Answer Compromise Solutions: Minimize distance to ideal point (w.r.t. some metric). f 2 ( x ) f 1 ( x ) Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Introduction 1 Definitions and Notations 2 Basic Properties 3 Approximation 4 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Introduction 1 Definitions and Notations 2 Basic Properties 3 Approximation 4 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Definition (Ideal Point) Given a multicriteria optimization problem max y ∈Y y , the ideal point y ∗ = ( y ∗ 1 , . . . , y ∗ k ) is defined by y ∗ i = max y ∈Y y i ∀ i. y 2 y ∗ Y y 1 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem max y ∈Y y with the ideal point y ∗ ∈ ◗ k , the Compromise Solution w.r.t. the norm �·� on ◗ k is y CS = min y ∈Y � y ∗ − y � . y 2 y ∗ y 1 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Definition (Compromise Solution, Yu 1973) Given a multicriteria optimization problem max y ∈Y y with the ideal point y ∗ ∈ ◗ k , the Compromise Solution w.r.t. the norm �·� on ◗ k is y CS = min y ∈Y � y ∗ − y � . y 2 y ∗ y 1 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation The norms we consider: � k � 1 / p � y p ( ℓ p -Norm) � y � p := , p ∈ [1 , ∞ ) i i =1 (Maximum ( ℓ ∞ -)Norm) � y � ∞ := max i =1 ,...,k y i | p := � y � ∞ + 1 | | | y | | p � y � 1 , p ∈ [1 , ∞ ] ( Cornered p -Norm) 1 1 p = 1 p = 2 p = 5 1 1 p = ∞ ℓ p -Norm Cornered p -Norm Degree of balancing controlled by adjusting p . Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation The norms we consider: � k � 1 / p � y p ( ℓ p -Norm) � y � p := , p ∈ [1 , ∞ ) i i =1 (Maximum ( ℓ ∞ -)Norm) � y � ∞ := max i =1 ,...,k y i | p := � y � ∞ + 1 | | | y | | p � y � 1 , p ∈ [1 , ∞ ] ( Cornered p -Norm) Weighted version: For any norm and λ ∈ ◗ k , λ ≥ 0 , λ � = 0 : � y � λ = � ( λ 1 y 1 , λ 2 y 2 , . . . , λ k y k ) � . Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Introduction 1 Definitions and Notations 2 Basic Properties 3 Approximation 4 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Known Properties Notation: For a family of norms �·� p , p ∈ [1 , ∞ ] , define CS ( λ, p ) := { Compromise Solution w.r.t. �·� λ p } CS ( p ) := { CS ( λ, p ) : λ ∈ ◗ k , λ ≥ 0 , λ � = 0 } Gearhardt 1979: Pareto optimality: CS ( p ) ⊆ Y P for p < ∞ . Y discrete ⇒ Y P = CS ( p ) for p big enough. y 2 Y P y 1 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Known Properties Notation: For a family of norms �·� p , p ∈ [1 , ∞ ] , define CS ( λ, p ) := { Compromise Solution w.r.t. �·� λ p } CS ( p ) := { CS ( λ, p ) : λ ∈ ◗ k , λ ≥ 0 , λ � = 0 } Gearhardt 1979: Pareto optimality: CS ( p ) ⊆ Y P for p < ∞ . Y discrete ⇒ Y P = CS ( p ) for p big enough. How big? y 2 Y P y 1 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation How to choose p Consider the Cornered Norm | | | · | | | p . Assumptions: Integrality: Y ⊂ ◆ k 0 Boundedness: ∃ polynomial π : y i ≤ 2 π ( | I | ) ∀ y ∈ Y , i = 1 , . . . , k Lemma (B¨ using,G.,Matuschke) Let M := 2 π ( | I | ) . For p > kM , it holds that Y P = CS ( p ) . Remarks: p can be chosen such that it has polynomial encoding length. ℓ p -norm: p � M log k suffices. Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Introduction 1 Definitions and Notations 2 Basic Properties 3 Approximation 4 Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Another way to cope with exponential size of Y P : Definition ( ε -approximate Pareto set) Let Y P be the Pareto set of a given instance, and let ε > 0 . Y εP ⊆ Y is an ε -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y εP such that y i ≤ (1 + ε ) y ′ ∀ i = 1 , . . . , k i Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Another way to cope with exponential size of Y P : Definition ( ε -approximate Pareto set) Let Y P be the Pareto set of a given instance, and let ε > 0 . Y εP ⊆ Y is an ε -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y εP such that y i ≤ (1 + ε ) y ′ ∀ i = 1 , . . . , k i Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Another way to cope with exponential size of Y P : Definition ( ε -approximate Pareto set) Let Y P be the Pareto set of a given instance, and let ε > 0 . Y εP ⊆ Y is an ε -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y εP such that y i ≤ (1 + ε ) y ′ ∀ i = 1 , . . . , k i Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Another way to cope with exponential size of Y P : Definition ( ε -approximate Pareto set) Let Y P be the Pareto set of a given instance, and let ε > 0 . Y εP ⊆ Y is an ε -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y εP such that y i ≤ (1 + ε ) y ′ ∀ i = 1 , . . . , k i Kai-Simon Goetzmann Compromise Solutions

Introduction Definitions and Notations Basic Properties Approximation Another way to cope with exponential size of Y P : Definition ( ε -approximate Pareto set) Let Y P be the Pareto set of a given instance, and let ε > 0 . Y εP ⊆ Y is an ε -approximate Pareto set if for all y ∈ Y P there is y ′ ∈ Y εP such that y i ≤ (1 + ε ) y ′ ∀ i = 1 , . . . , k i Theorem (Papadimitriou&Yannakakis,2000) There always exists an ε -approximate Pareto set with size polynomial in | I | and 1 / ε . Kai-Simon Goetzmann Compromise Solutions